Which vector is equivalent to 𝐮 plus 𝐯?
Looking at our figure, we see these two vectors making up the sides of a four-sided shape. Since the length of this side equals the magnitude of vector 𝐯 and so does the length of this side, while these two sides of our shape have lengths equal to the magnitude of vector 𝐮, we know the shape overall is a parallelogram. Using this shape then, there are actually two ways to find our answer.
Starting at point 𝐴, we could follow along vector 𝐮 and then after that vector 𝐯, and the tip of vector 𝐯 points to where the sum of these two vectors lies. Here we’re using what’s called the tip-to-tail method. Note that the tail of vector 𝐯 lies at of the tip of vector 𝐮. Arranged this way, the sum of these two vectors 𝐮 plus 𝐯 is indicated by a vector that goes from the tail of the first vector, vector 𝐮, to the tip of the second vector, vector 𝐯.
We mentioned that there’s a second way to solve for 𝐮 plus 𝐯. And our shape now shows us how. 𝐮 plus 𝐯 is equivalent to 𝐯 plus 𝐮. So we could get the same result by starting at the tail of vector 𝐯 and then where its tip meets the tail of vector 𝐮, following that vector to its end. This approach gives us the same result of vector 𝐮 plus vector 𝐯. In terms of the corners of our parallelogram, we see that this resultant vector joins point 𝐴 to point 𝐷. We can express a vector from point 𝐴 to point 𝐷 like this. This, then, is our final answer. Vector 𝐮 plus vector 𝐯 equals vector 𝐀𝐃.